Virtual worlds, construed in a broad enough sense to include text-based systems, as well as video games, new media, augmented reality, and user interfaces of all kinds, are increasingly important in scientific research, entertainment, communication, commerce, and art. However, we lack scientific theories that can adequately support the design of such virtual worlds, even in simple cases. Semiotics would seem a natural source for such theories, but this field lacks the precision needed for engineering applications, and also fails to addresses interaction and social issues, both of which are crucial for applications to communication and collaboration. This chapter suggests an approach called algebraic semiotics to help solve these and related problems, by providing precise application-oriented basic concepts such as sign, representation, and representation quality, and a calculus of representation that includes blending. This chapter also includes some theory for narrative and metaphor, and case studies on information visualization, proof presentation, humor, and user interaction.